Simplify the radical. [tex]$\sqrt[7]{(x-6)^7}$[/tex]
Answer (1)
The problem asks to simplify the radical expression 7 ( x − 6 ) 7 .
Recognize that the 7th root and 7th power are inverse operations.
Simplify the expression directly, preserving the sign since it's an odd root.
The simplified expression is x − 6 .
Explanation
Understanding the Problem We are asked to simplify the radical expression 7 ( x − 6 ) 7 . This involves understanding how radicals and exponents interact, especially when dealing with odd roots.
Simplifying the Expression The key to simplifying this expression lies in recognizing that the 7th root and the 7th power are inverse operations. When we take the 7th root of something raised to the 7th power, we essentially undo the exponentiation. However, we need to be mindful of the sign of the base, ( x − 6 ) , especially when dealing with even roots. But since we have an odd root (7th root), the sign will be preserved.
Final Simplification Since we are taking an odd root (7th root), we don't need to worry about absolute values. The expression simplifies directly to x − 6 .
Conclusion Therefore, the simplified form of the given radical expression is x − 6 .
Examples
Imagine you are designing a box-shaped container where the volume is expressed as ( x − 6 ) 7 cubic units. To find the length of one side of the box, you would need to take the 7th root of the volume, which simplifies to 7 ( x − 6 ) 7 = x − 6 . This tells you that the side length of the box is x − 6 units. Understanding how to simplify radicals helps in determining dimensions from volumes or vice versa in various practical scenarios.
